Jean-Christophe Jameux
4 years ago
parent
67cb5b0241
commit
e849db9ded
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#LyX 2.3 created this file. For more info see http://www.lyx.org/ |
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Étudier sur |
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la fonction |
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et la représenter graphiquement dans le cadre ci-dessous. |
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On fera apparaitre dans ce cadre le détail des calculs, en particulier de |
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la dérivée, et on donnera son tableau de variations en faisant apparaitre |
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les limites éventuelles. |
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Représenter graphiquement sur |
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la fonction |
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dans ce cadre. |
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@ -0,0 +1,354 @@ |
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|
#LyX 2.3 created this file. For more info see http://www.lyx.org/ |
||||||
|
\lyxformat 544 |
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Soit la suite |
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\begin_inset Formula $u$ |
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\end_inset |
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telle que |
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\begin_inset Formula $u_{0}=2$ |
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\end_inset |
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et |
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\begin_inset Formula $u_{n+1}=5u_{n}+4$ |
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\end_inset |
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pour tout entier |
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, et soit |
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\begin_inset Formula $v$ |
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\end_inset |
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la suite définie par |
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\begin_inset Formula $v_{n}=u_{n}+1$ |
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\end_inset |
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pour tout entier |
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\begin_inset Formula $n\geqslant1$ |
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\end_inset |
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. |
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Montrer que la suite |
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\begin_inset Formula $v$ |
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\end_inset |
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est une suite géométrique dont on donnera la raison et le premier terme |
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puis donner une formule pour |
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\begin_inset Formula $v_{n}$ |
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\end_inset |
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puis pour |
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\begin_inset Formula $u_{n}$ |
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\end_inset |
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en fonction de l'entier naturel |
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\begin_inset Formula $n$ |
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\end_inset |
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. |
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Établir, en faisant apparaitre les calculs, le tableau de signe sur |
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\end_inset |
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de la fonction |
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et en déduire les solutions sur |
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de l'inéquation |
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Binary file not shown.
@ -0,0 +1,112 @@ |
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a : 2,3,4,5,6,7 |
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b : 2,3,4,5,6,7 |
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c : 2,3,4,5,6,7 |
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\subsection*{Exercice 1} |
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Pour chacune des propositions de cet exercice, cochez A si elle est vraie\par et B si elle est fausse. |
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On s'intéresse à la fonction f définie par $f:x\mapsto\left(#a#x+#b#\right)e^{#c#x}$. |
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vrfx |
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+ L'ensemble de définition de f est $\left]-\infty,+\infty\right[$. |
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- L'ensemble de définition de f est $\left]0,+\infty\right[$. |
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vrfx |
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+ La dérivée de f est donnée par $f^{\prime}:x\mapsto\left(#a*c#x+#a+b*c#\right)e^{#c#x}$. |
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|
- La dérivée de f est donnée par $f^{\prime}:x\mapsto\left(#a*c#\right)e^{cx}$. |
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||||||
|
On admet que son approximation quadratique est donné par\par $\qquad f\left(x\right)\approx #b# + #a*(b + 1)# x + #1/2 *a^2 *(b + 2)# x^2$. |
||||||
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||||||
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vrfx |
||||||
|
+ L'équation de la tangente $T$ à la courbe représentative $\mathscr C$ de $f$ en son point d'abscisse $0$ est donnée par $y=#a* (b + 1)# x +#b# $. |
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||||||
|
- L'équation de la tangente $T$ à la courbe représentative $\mathscr C$ de $f$ en son point d'abscisse $0$ est donnée par $y=#b#x + #a*(b + 1)# $. |
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vrfx |
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+ $T$ est au-dessous de $\mathscr C$. |
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\subsection*{Exercice 2} |
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|
Cet exercice doit être rédigé au verso de la grille réponse. |
||||||
|
\subsection*{Exercice 3} |
||||||
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|
||||||
|
qcm Parmi ces fonctions, laquelle est une primitive de la fonction $f$\par définie par $ f(x)=x \cdot \cos (3 x+1)$ ? |
||||||
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|
||||||
|
- $F:x \mapsto \cos (3 x+1)-3 x \cdot \sin (3 x+1)$ |
||||||
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||||||
|
- $G:x \mapsto x \cdot \sin (3 x+1)$ |
||||||
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||||||
|
+ $H:x \mapsto \frac{x \cdot \sin (3 x+1)}{3}+\frac{\cos (3 x+1)}{9}$ |
||||||
|
|
||||||
|
|
||||||
|
qcm $\int\limits_{0}^{2}\left(5 t^{3}+3 t^{2}-t+7\right) d t=$ |
||||||
|
|
||||||
|
+ 40 |
||||||
|
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||||||
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- 71 |
||||||
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||||||
|
- $15 t^{2}+6 t-1$ |
||||||
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||||||
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||||||
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|
||||||
|
qcm La valeur moyenne de la fonction $g: x \rightarrow e^{5 x} $ entre 0 et 2 est de : |
||||||
|
|
||||||
|
-environ $22 025,5$ |
||||||
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|
||||||
|
-environ $4405,1$ |
||||||
|
|
||||||
|
+environ $2 202,5$ |
||||||
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||||||
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||||||
|
p : 2,3,4 |
||||||
|
a : 3,4,5 |
||||||
|
b : 10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30 |
||||||
|
\newpage |
||||||
|
\subsection*{Problème} |
||||||
|
Alain Raffletou cherche à connaitre le nombre moyen de clients journaliers dans une de ses boutiques sur l'année en cours. Pour cela, il modélise le nombre de clients par une fonction donnant le nombre de clients $c\left(t\right)$ en fonction du temps $t$, exprimé en jours. On prends donc $t\in\left[0;365\right]$. |
||||||
|
|
||||||
|
qcm En remarquant que le nombre de clients ne peut être négatif, déterminé, parmi les propositions ci-dessous, laquelle donne la formule de $c\left(t\right)$ en fonction de $t$ : |
||||||
|
|
||||||
|
+$c\left(t\right)=#a#\cos\left(\frac{#2*p#\pi t}{365}\right)+#b#$ |
||||||
|
|
||||||
|
-$c\left(t\right)=#a#\sin\left(\frac{#2*p#\pi t}{365}\right)-#b#$ |
||||||
|
|
||||||
|
-$c\left(t\right)=#a#\sin\left(\frac{#2*p#\pi t}{365}+#b#\right)$ |
||||||
|
|
||||||
|
-$c\left(t\right)=#a#\cos\left(\frac{#2*p#\pi t}{365}-#b#\right)$ |
||||||
|
|
||||||
|
qcm On a alors que les valeurs de $c\left(t\right)$ sont comprises entre : |
||||||
|
|
||||||
|
+$#b-a#$ et $#b+a#$ |
||||||
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||||||
|
-$0$ et $#a#$ |
||||||
|
|
||||||
|
-$0$ et $#b#$ |
||||||
|
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||||||
|
-$0$ et $#b+a#$ |
||||||
|
|
||||||
|
Dès lors, il peut calculer le nombre moyen de clients en appliquant la formule |
||||||
|
$\frac{1}{365}\int\limits _{0}^{365}C\left(t\right)dt$ , où $C$ désigne une primitive de $c$. |
||||||
|
|
||||||
|
qcm Parmi les propositions ci-dessous, laquelle donne une formule possible C~? |
||||||
|
|
||||||
|
-$C\left(t\right)=-#a#\sin\left(\frac{#2*p#\pi t}{365}\right)$ |
||||||
|
|
||||||
|
-$C\left(t\right)=\frac{#2*a*p#}{365}\pi\sin\left(\frac{#2*p#\pi t}{365}\right)-#b#t+#b-2*a+1#$ |
||||||
|
|
||||||
|
-$C\left(t\right)=\frac{#2*a*p#}{365}\pi\cos\left(\frac{#2*p#\pi t}{365}\right)-#b#t-#2*b-3*a#$ |
||||||
|
|
||||||
|
+$C\left(t\right)=\frac{#365*a#}{#2p#}\pi\sin\left(\frac{#2*p#\pi t}{365}\right)+#b#t$ |
||||||
|
|
||||||
|
qcm On obtient ainsi que le nombre moyen de clients journaliers sur l'année en cours est donné par : |
||||||
|
|
||||||
|
-$#a#$ |
||||||
|
|
||||||
|
+$#b#$ |
||||||
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||||||
|
-$#a#\cos\left(#2*p#\pi\right)$ |
||||||
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||||||
|
-$#b#\sin\left(#2*p#\pi\right)$ |
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("C:\Program Files\MiKTeX\tex/context/base/mkii\supp-pdf.mkii" |
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[Loading MPS to PDF converter (version 2006.09.02).] |
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\scratchcounter=\count268 |
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\scratchdimen=\dimen156 |
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\scratchbox=\box50 |
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\nofMPsegments=\count269 |
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\nofMParguments=\count270 |
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\everyMPshowfont=\toks24 |
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\MPscratchCnt=\count271 |
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\MPscratchDim=\dimen157 |
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\MPnumerator=\count272 |
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\makeMPintoPDFobject=\count273 |
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\everyMPtoPDFconversion=\toks25 |
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) |
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LaTeX Font Info: Trying to load font information for U+msa on input line 59. |
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|
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("C:\Program Files\MiKTeX\tex/latex/amsfonts\umsa.fd" |
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File: umsa.fd 2013/01/14 v3.01 AMS symbols A |
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) |
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LaTeX Font Info: Trying to load font information for U+msb on input line 59. |
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|
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|
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("C:\Program Files\MiKTeX\tex/latex/amsfonts\umsb.fd" |
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File: umsb.fd 2013/01/14 v3.01 AMS symbols B |
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|
) [1 |
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|
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{C:/Users/Echophile/AppData/Local/MiKTeX/pdftex/config/pdftex.map}] [2] [3] [4] |
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|
[5] [6] [7] [8] |
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|
[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] |
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|
[24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] |
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|
[39] [40] [41] [42] [43] [44] [45 |
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|
|
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|
] ("BTS 2- DST 2 - Sujets.aux") ) |
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|
Here is how much of TeX's memory you used: |
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|
3294 strings out of 479799 |
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|
41865 string characters out of 2879534 |
||||||
|
326797 words of memory out of 3000000 |
||||||
|
20286 multiletter control sequences out of 15000+200000 |
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|
539093 words of font info for 40 fonts, out of 3000000 for 9000 |
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|
1141 hyphenation exceptions out of 8191 |
||||||
|
66i,5n,72p,241b,204s stack positions out of 5000i,500n,10000p,200000b,50000s |
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|
<C:\Users\Echophile\AppData\Local\MiKTeX\font |
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|
s/pk/ljfour/jknappen/ec/dpi600\ecrm1000.pk> <C:\Users\Echophile\AppData\Local\M |
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|
iKTeX\fonts/pk/ljfour/jknappen/ec/dpi600\ecbx1440.pk><C:/Program Files/MiKTeX/f |
||||||
|
onts/type1/public/amsfonts/cm/cmmi10.pfb><C:/Program Files/MiKTeX/fonts/type1/p |
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|
ublic/amsfonts/cm/cmr10.pfb><C:/Program Files/MiKTeX/fonts/type1/public/amsfont |
||||||
|
s/cm/cmr7.pfb><C:/Program Files/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy10.pf |
||||||
|
b> |
||||||
|
Output written on "BTS 2- DST 2 - Sujets.pdf" (45 pages, 99233 bytes). |
||||||
|
PDF statistics: |
||||||
|
229 PDF objects out of 1000 (max. 8388607) |
||||||
|
0 named destinations out of 1000 (max. 500000) |
||||||
|
1 words of extra memory for PDF output out of 10000 (max. 10000000) |
||||||
|
|
||||||
@ -0,0 +1,19 @@ |
|||||||
|
This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 20.11) (preloaded format=pdflatex 2020.11.23) 8 NOV 2021 15:53 |
||||||
|
entering extended mode |
||||||
|
**.\BTS 2- DST 2 - Sujets.tex |
||||||
|
|
||||||
|
! Emergency stop. |
||||||
|
<*> .\BTS |
||||||
|
2- DST 2 - Sujets.tex |
||||||
|
End of file on the terminal! |
||||||
|
|
||||||
|
|
||||||
|
Here is how much of TeX's memory you used: |
||||||
|
0 strings out of 479799 |
||||||
|
1 string characters out of 2879534 |
||||||
|
264509 words of memory out of 3000000 |
||||||
|
17099 multiletter control sequences out of 15000+200000 |
||||||
|
535088 words of font info for 29 fonts, out of 3000000 for 9000 |
||||||
|
1141 hyphenation exceptions out of 8191 |
||||||
|
0i,0n,0p,1b,6s stack positions out of 5000i,500n,10000p,200000b,50000s |
||||||
|
! ==> Fatal error occurred, no output PDF file produced! |
||||||
Binary file not shown.
Binary file not shown.
Loading…
Reference in new issue